Hub Location as the Minimization of a Supermodular Set Function

This paper highlights how a general class of hub location problems can be modeled as the minimization of a real-valued supermodular set function. Well-known problems such as uncapacitated hub location, p -hub median, and hub arc location, among others, are shown to be particular cases of this class. Two integer programming formulations are introduced and compared. One uses path-based variables, frequently employed in hub location, whereas the other exploits properties of supermodular functions. In addition, several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised. Computational experiments are performed to compare both formulations when used with a general purpose solver. Computational results obtained on benchmark instances confirm the superiority of the supermodular formulation.

[1]  L. Wolsey.  Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems , 1982, Math. Oper. Res..

[2]  George L. Nemhauser,et al.  The uncapacitated facility location problem , 1990 .

[3]  Iván A. Contreras,et al.  The Tree of Hubs Location Problem , 2010, Eur. J. Oper. Res..

[4]  M. O'Kelly,et al.  A quadratic integer program for the location of interacting hub facilities , 1987 .

[5]  Hande Yaman,et al.  Star p-hub median problem with modular arc capacities , 2008, Comput. Oper. Res..

[6]  Boris Goldengorin.  Maximization of submodular functions: Theory and enumeration algorithms , 2009, Eur. J. Oper. Res..

[7]  George L. Nemhauser,et al.  Note--On "Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms" , 1979 .

[8]  A. M. FRIEZE.  A cost function property for plant location problems , 1974, Math. Program..

[9]  James F. Campbell,et al.  Integer programming formulations of discrete hub location problems , 1994 .

[10]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[11]  Laurence A. Wolsey,et al.  Best Algorithms for Approximating the Maximum of a Submodular Set Function , 1978, Math. Oper. Res..

[12]  Andreas T. Ernst,et al.  Preprocessing and cutting for multiple allocation hub location problems , 2004, Eur. J. Oper. Res..

[13]  Mercedes Landete,et al.  New formulations for the uncapacitated multiple allocation hub location problem , 2006, Eur. J. Oper. Res..

[14]  Morton E. O'Kelly,et al.  Twenty-Five Years of Hub Location Research , 2012, Transp. Sci..

[15]  G. Nemhauser,et al.  On the Uncapacitated Location Problem , 1977 .

[16]  Stefan Nickel,et al.  Hub location problems in transportation networks , 2011 .

[17]  Iván A. Contreras,et al.  Tight bounds from a path based formulation for the tree of hub location problem , 2009, Comput. Oper. Res..

[18]  Andreas T. Ernst,et al.  Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem , 1998 .

[19]  Gilbert Laporte,et al.  Exact Solution of Large-Scale Hub Location Problems with Multiple Capacity Levels , 2012, Transp. Sci..

[20]  John R. Current,et al.  The hub location and network design problem with fixed and variable arc costs: formulation and dual-based solution heuristic , 2008, J. Oper. Res. Soc..

[21]  Harvey J. Miller,et al.  THE HUB NETWORK DESIGN PROBLEM: A REVIEW AND SYNTHESIS. , 1994 .

[22]  Andreas T. Ernst,et al.  Hub Arc Location Problems: Part II - Formulations and Optimal Algorithms , 2005, Manag. Sci..

[23]  Gilbert Laporte,et al.  Benders Decomposition for Large-Scale Uncapacitated Hub Location , 2011, Oper. Res..

[24]  G. Nemhauser,et al.  Maximizing Submodular Set Functions: Formulations and Analysis of Algorithms* , 1981 .

[25]  Oded Berman,et al.  Locating Discretionary Service Facilities, II: Maximizing Market Size, Minimizing Inconvenience , 1995, Oper. Res..

[26]  Stefan Nickel,et al.  Hub Location Problems in Urban Traffic Networks , 2001 .

[27]  Andreas T. Ernst,et al.  An Exact Solution Approach Based on Shortest-Paths for p-Hub Median Problems , 1998, INFORMS J. Comput..

[28]  Alfredo Marín,et al.  Uncapacitated Euclidean Hub Location: Strengthened Formulation, New Facets and a Relax-and-cut Algorithm , 2005, J. Glob. Optim..

[29]  Morton E. O'Kelly,et al.  Hub facility location with fixed costs , 1992 .

[30]  Andreas T. Ernst,et al.  Hub location problems , 2002 .

[31]  Horst W. Hamacher,et al.  Adapting polyhedral properties from facility to hub location problems , 2004, Discret. Appl. Math..

[32]  Iván A. Contreras,et al.  General network design: A unified view of combined location and network design problems , 2012, Eur. J. Oper. Res..

[33]  Sibel A. Alumur,et al.  The design of single allocation incomplete hub networks , 2009 .

[34]  Morton E. O'Kelly,et al.  The Location of Interacting Hub Facilities , 1986, Transp. Sci..

[35]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[36]  M. O'Kelly,et al.  Activity Levels at Hub Facilities in Interacting Networks , 2010 .

[37]  Mercedes Landete,et al.  New formulation and a branch-and-cut algorithm for the multiple allocation p-hub median problem , 2012, Eur. J. Oper. Res..

[38]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

[39]  Iván A. Contreras,et al.  Branch and Price for Large-Scale Capacitated Hub Location Problems with Single Assignment , 2011, INFORMS J. Comput..

[40]  Martine Labbé,et al.  A branch and cut algorithm for hub location problems with single assignment , 2005, Math. Program..

[41]  J. G. Klincewicz,et al.  HUB LOCATION IN BACKBONE/TRIBUTARY NETWORK DESIGN: A REVIEW , 1998 .

[42]  M. Labbé,et al.  Solving the hub location problem in a star–star network , 2008 .

[43]  Gérard Cornuéjols,et al.  Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem , 1984, Discret. Appl. Math..

[44]  Jeff A. Bilmes,et al.  Submodularity Cuts and Applications , 2009, NIPS.

[45]  Andreas T. Ernst,et al.  Hub Arc Location Problems: Part I - Introduction and Results , 2005, Manag. Sci..

[46]  Sibel A. Alumur,et al.  Network hub location problems: The state of the art , 2008, Eur. J. Oper. Res..

[47]  James F. Campbell.  Hub Location and the p-Hub Median Problem , 1996, Oper. Res..

[48]  G. Nemhauser,et al.  Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case☆ , 1996 .